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 PDE Seminar

For further information, or to suggest a PDE Seminar speaker, please contact:

Misha Perepelitsa Gabriela Jaramillo William Fitzgibbon


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Seminar on Partial Differential Equations
Spring 2023

Friday at 2:00 PM
Click here for previous seminars of Fall 2022

February 3
February 10  
 February 17  
February 24  
March 3 Dr. Chris Henderson (University of Arizona)

Title: The shape defect function and stability of traveling waves

In their original paper, Kolmogorov, Petrovsky, and Piskunov demonstrated stability of the minimal speed traveling wave with an ingenious argument based on, roughly, the decreasing steepness of the profile. This proof is extremely flexible, yet entirely not quantitative being based on compactness. On the other hand, more modern PDE proofs of the stability of traveling waves solutions to reaction-diffusion equations are highly tailored to the particular equation, fairly complicated, and often not sharp in terms of the rate of convergence. In this talk, I will introduce a natural quantity, the shape defect function, that allows a simple approach to quantifying convergence to the traveling wave for a large class of reaction-diffusion equations, including both pushed, pulled, and pushmi-pullyu equations. This is a joint work with Jing An and Lenya Ryzhik.

March 10 Dr. Tien Khai Nguyen (NC State University)

Title: The metric entropy for nonlinear PDEs

Inspired by a conjecture posed by Lax in 2002, in recent years it has received increasing attention to the study on the metric entropy for nonlinear PDEs. In this talk, I will present recent results on sharp estimates in terms of epsilon entropy for hyperbolic conservation laws and Hamilton-Jacobi equations. Estimates of this type play a central role in various areas of information theory and statistics as well as ergodic and learning theory. In the present setting, this concept could provide a measure of the order of "resolution" of a numerical method for the corresponding equations.

March 17

March 24  
March 31 (12pm) Dr. Laura Kanzler (Universite Paris-Dauphine)

Title: Kinetic Modelling of Colonies of Myxobacteria

Abstract: Myxobacteria are rod-shaped, social bacteria that are able to move on a surface by `gliding', and form a fascinating example of how simple cell-cell interaction rules can lead to emergent, collective behavior. In this talk a new kinetic model of Boltzmann-type for such colonies of myxobacteria will be formally derived and investigated. For the spatially homogeneous case an existence and uniqueness result will be shown, as well as exponential decay to an equilibrium for the Maxwellian collision operator. Furthermore, a model extension including Brownian forcing in velocity direction during the free flight phase of bacteria as well as insights in its asymptotic behavior will be presented.

The methods used for the analysis combine several tools from kinetic theory, entropy methods, hypocoercivity as well as optimal transport. The talk will be concluded with numerical simulations for the spatially homogeneous case, which are confirming the analytical results.

April 7  
April 14 Dr. Cinzia Soresina (Karl-Franzens-University of Graz)

Title: Cross-diffusion systems in population dynamics: derivation, bifurcations and patter formation

Abstract: In population dynamics, cross-diffusion describes the influence of one species on the diffusion of another. The cross-diffusion SKT model was proposed to account for stable inhomogeneous steady states exhibiting spatial segregation of two competing species. Even though the reaction part does not present the activator-inhibitor structure, the cross-diffusion terms are the key ingredient for the appearance of spatial patterns. We provide a deeper understanding of the conditions required for non-homogeneous steady states to exist, focusing on multistability regions and on the presence of time-periodic spatial patterns, by combining a detailed linearised and weakly non-linear analysis with advanced numerical bifurcation methods via the continuation software pde2path [1,2].

Even though the particular form of cross-diffusion terms in the SKT model may seem artificial, they naturally incorporates processes occurring at different time scales. It can be easily seen, at least at a formal level, that cross-diffusion appears in the fast-reaction limit of a "microscopic" model (in terms of time scales) presenting only standard diffusion and fast-reaction terms. The same approach can also be exploited in other contexts, e.g. predator-prey interactions, plant ecology and epidemiology.

[1] M. Breden, C. Kuehn, C. Soresina, On the influence of cross-diffusion in pattern formation, Journal of Computational Dynamics, 8(2):21, 2021.
[2] C. Soresina, Hopf bifurcations in the full SKT model and where to find them, Discrete and Continuous Dynamical Systems - S, 15(9):2673-2693, 2022.

April 21

April 28 Dr. Paul Carter (UC Irvine)

Title: Instability of planar interfaces in reaction-diffusion-advection equations

Abstract: We consider planar interfaces between stable homogeneous rest states in singularly perturbed 2-component reaction diffusion advection equations, motivated by the appearance of fronts between bare soil and vegetation in dryland ecosystems, as well as multi-interface solutions, such as vegetation stripes. On sloped terrain, one can find stable traveling interfaces, while on flat ground, one finds that sideband instabilities along the interface can lead to labyrinthine Turing-like patterns. To explore this behavior, using geometric singular perturbation methods, we analyze instability criteria for planar interfaces in reaction diffusion advection systems, focussing on a specific Klausmeier-type model, and examine the effect of terrain slope on the stability of the interfaces.

Small changes/shifts in the dates may be possible.

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